Computer-aided molecular design: An introduction and review of tools, applications, and solution techniques

0.86 MB·

by  Nick D. Austin

Computer-aided molecular design: An
introduction and review of tools, applications,
and solution techniques
7 Nick D. Austin1, Nikolaos V. Sahinidis1, and Daniel W. Trahan2
1
0
2 1Carnegie Mellon University, Pittsburgh, PA, USA

n 2The Dow Chemical Company, Freeport, TX, USA
a
J

5
1
Abstract

]
E
Thisarticleprovidesanintroductiontoandreviewofthefieldofcomputer-aidedmoleculardesign(CAMD).
C
It is intended to be approachable for the absolute beginner as well as useful to the seasoned CAMD
.
s practitioner. We begin by discussing various quantitative structure-property relationships (QSPRs) which
c
have been demonstrated to work well with CAMD problems. The methods discussed in this article are
[
(1) group contribution methods, (2) topological indices, and (3) signature descriptors. Next, we present

1 generaloptimization formulations for various forms of the CAMD problem. Common designconstraints are
v
discussedandstructuralfeasibilityconstraintsareprovidedforthethreetypesofQSPRsaddressed. Wethen
8
detail useful techniques for approaching CAMD optimization problems, including decomposition methods,
7
9 heuristic approaches,andmathematicalprogrammingstrategies. Finally,we discuss many applicationsthat
3 have been addressed using CAMD.
0 Keywords: Computer-aided mixture design; Computer-aided molecular design; Integrated product and
.
1 process design; Group contribution; Topological indices; Signature descriptors
0
7
1
1 Introduction The process of determining new and suitable
:
v chemicals for a certain application can be generally
i
X The application of chemistry to manipulate the termed chemical product design [34]. Chemical
naturalworldhasitsearliestexamplesinmetalwork- product design has long been a laborious, trial-and-
r
a ing and pottery [134], with some pottery artifacts error procedure, limited often by a fixed amount of
reportedto be asoldas20,000years[167]. Chemical chemical,time,andfinancialresources. Designefforts
products have since played an important role in are often high-throughput and tend to focus on a
history and left an indelible mark on the way small class of compounds or structural analogues of
we live and work. Early history included basic known chemicals. Accordingly, the so-called “design
incendiary fuels, perfumes, and soap as some of the space”—the set of unique molecular structures
first widespread uses of chemicals. The modern considered—is often quite small for these product
age has witnessed an unprecedented expansion design problems, especially considering the massive
of chemical products, including pesticides, fuels design space of all possible chemical structures.
for transportation and electricity, pharmaceuticals, It is then clear that to keep pace with the
plastics,andabroadarrayofindustrialandconsumer growing demand for new chemical products and to
products. True to these historical trends, few things adequately explore the full chemical design space,
areaspervasiveas chemicalproductsin21stcentury other approaches must be considered. Fortunately,
life, and there are an ever-increasing number of the availability and efficiency of computational
new chemical applications which require specialized resources makes these design problems more tenable
compounds. than ever before. Noteworthy among computational
c2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
(cid:13)
http://creativecommons.org/licenses/by-nc-nd/4.0/. The formal publication of this article
is available at http://dx.doi.org/10.1016/j.cherd.2016.10.014.
CAMD Review 2
approaches is the field of computer-aided molecular ensure practical solutions as well as the chemical
design (CAMD), which leverages the simplicity feasibility of the designed structures. In Section 4,
of semi-empirical quantitative structure-property various solution techniques for the CAMD problem
relationships (QSPRs) in conjunction with fast and are discussed, including mathematical optimization
efficient numerical optimization algorithms. strategies, decomposition methods, and heuristic
CAMD has its roots in the 1980’s, although the approaches. Finally, in Section 5, we provide a
general use of computers in chemistry pre-dates this diversethoughnon-exhaustive reviewof applications
by a few decades. Stated formally, the CAMD of CAMD problems.
problem concerns designing an optimal molecular
structure(s) for a certain application. CAMD
2 Popular types of QSPRs in CAMD
combines molecular modeling techniques, thermody-
namics, and numerical optimization to design good
The CAMD problem attempts to choose optimal (or
or optimal molecular structures, many of them often
simply good) molecules for some purpose from the
completely novel. Advances in chemical modeling in
space of theoretically possible chemical structures.
the last few decades have greatly benefited CAMD,
At first glance, the CAMD problem must consider a
andpractitionersarenowcapableofrelatingchemical
veryabstractchemicaldesignspace of atoms,bonds,
structures to properties at several levels of accuracy
aromaticity,structuralisomers,electroniceffects,etc.
(molecular mechanics, semi-empirical, ab initio).
Though many of these features are certainly what
Though CAMD often uses semi-empirical modeling
givesmoleculestheir specificpropertiesandchemical
techniques for their simplicity and efficiency, new
functionality,they aredifficultto buildinto anytype
approaches incorporating more accurate methods
of optimization scheme. This is primarily because
are emerging. Modern combinatorial optimization
there is no available immediate relationship between
techniques are also essential for CAMD, enabling
an arbitrary chemical structure and its performance
the optimization over staggeringly large design
or suitability regarding a specific application. In
spaces which would otherwise be inaccessible (using
order to “rank” different structures and choose an
enumeration algorithms, for example).
optimal one, we must have some efficient way to
This article is intended to provide a description
quantify the properties and performance of each
of popular QSPRs used in CAMD, the CAMD
structure.
problem itself, and several solution approaches to
A second issue is the sheer size of the chemical
its various forms. There exist several CAMD
search space. At the time of writing this article,
reviews [52, 126] that provide a thorough coverage
the CASregistry[27]reportsover115millionunique
of many relevant applications and milestones of
organic and inorganic structures. This number only
CAMD and are good resources for the interested
represents compounds which have been synthesized
reader. While also providing a review of CAMD
and cataloged, and it is already far too large
literature, the currentarticle first motivates the field
for every structure to be considered in any type
from the ground up. In this way, this review is
of trial-and-error design scheme. This number is
intended to be accessible to readers of all levels as it
also only a fraction of the theoretically possible
presents many essential CAMD tools and techniques
chemical space, which some estimates indicate
in detail. We also provide a discussion of various
may contain more than 1060 unique molecules for
techniques and a review of CAMD applications, so
small, drug-like structures [13]. Even with very
this document will also be of interest to the more
efficient ways to estimate the performance of a
advancedCAMDpractitioner. WebegininSection2
certain structure, screening these structures using
by detailing three popular classes of QSPRs which
an enumeration strategy is far beyond current
are often used in CAMD: (1) group contribution
computational capacity. For this reason, we also
methods; (2) topological indices; and (3) signature
needto relate the chemicalspace to a space that can
descriptors. Next, in Section 3, we present the
be utilized for combinatorial optimization, allowing
CAMD problem from a mathematical programming
us to design over the massive search space far more
perspective, discussing various classes of the single-
efficiently.
molecule design problem as well as CAMD problems
CAMDpractitionershavereliedonsemi-empirical
consideringmixturesofmoleculesandthoseinvolving
quantitative structure property relationships
the simultaneous design of a chemical product and
(QSPRs) to address both of these issues. First,
the process it is a part of. In this section,
many semi-empirical methods delineate a clear
several other important design considerations are
connection between the abstract chemical space and
presented, including a few important constraints to
Austin, Sahinidis, and Trahan 3
Figure 1: Propanol represented by its groups
molecules. To regress these parameters, the identity
of all of the groups must be specified a priori.
= CH Returning to our example, one can easily imagine
3
OH different sets of groups being used to describe
= CH
2 propanol. For example, the groups CH , CH ,
3 2
− − −
n-propanol = OH and CH2OH also completely accountfor the atoms
−
inpropanol,andthesemayprovideabetterfitforthe
regression problem. For this reason, different group
contributionmethodstoestimatedifferentproperties
usuallydonothaveconsistentsetsofgroups. Finally,
a more practical space of quantitative properties.
we note that the vector n is generally much bigger
These methods are also often simple and can be
as many group contribution methods contain 50-100
applied to estimate properties very efficiently.
groups. Many group contribution methods make the
Second, many of these methods break molecular
additional assumption that groups cannot overlap,
structure into sub-molecular collections of atoms
which means that n is typically a sparse vector. A
and bonds. These molecular sub-structures are
pictorial example of the usage of group contribution
assumed to dictate a molecule’s properties. Using
methods is given in Fig. 2. In this example, we
thesetypesofrepresentationsofthemolecularspace,
apply a hypothetical GC method to a hypothetical
combinatorial optimization can be directly applied
molecule. We show how a molecular structure is
to molecular design problems.
decomposed into its constituent groups and provide
a count of each of these groups. These counts
2.1 Group-contribution methods constitutethe elementsofthevectorn. Thenvector
ispairedwithahypotheticalcvector,andanexample
The most commonly used QSPRs in CAMD are
property is calculated.
group contribution (GC) methods. These work
One of the earliest examples of GC methods is
under the assumption that a molecule’s properties
from Benson and Buss [10], who are considered
can be predicted by the number of occurrences of
to be the originators of so-called “group increment
variousmolecularsub-structurescalled“groups.” For
theory.” Group increment theory, or Benson group
example, we may think to represent the simple
increment theory (BGIT), is analogous to GC
molecule propanol as a combination of the groups
methods, but these terms may be more common in
CH , CH , and OH. In this case, the dashes
− 3 − 2− − the physical chemistry literature. In the original
( ) represent bonds to other groups. In its group
− 1958 paper [10], Benson and Buss proposed a simple
representation,propanolwould no longer be thought
group additivity scheme for the prediction of bond
of as the connected alcohol molecule, but rather as
dissociation energies. Benson et al. [11] extended
some collection of its constituent groups. The group
this workto accountfor a greaterdiversity ofgroups
representation of propanol is shown in Fig. 1.
and to estimate heat capacities. Additional work
Being QSPRs, group contribution methods trans-
from Cohen and Benson includes estimating heats
latethegrouprepresentationofamolecularstructure
of formation with group increment theory [31]. A
into an estimate for some property P. To do this,
large number of additional efforts have used Benson-
group contribution methods define a vector n that
like increments to estimate the same thermophysical
represents the number of occurrences of each of the
properties, a very small sample of which are [36, 76,
groups. Assuming we only have the three groups
144].
shownabove,propanol’snvectorwouldlookliken=
Another very popular GC method was devised
[1,2,1],where the entries inthis vectorrepresentthe
by Joback and Reid [78]. This method extended
numberofoccurrencesofthegroups CH , CH ,
− 3 − 2− the group increment idea to model many different
and OH, respectively. Each of these groups g
− properties with the same set of groups. The
would also be associated with a coefficient c which
g Joback and Reid model also included functional
quantifies its affect or “contribution” to a particular
transformations for the original group increment
property P. Properties are calculated as follows:
summations. These altered the group contribution
definition to the following:
P = c n (1)
g g
g
X
P =f c n (2)
g g
The vector of coefficients c comes from regression !
g
of the property P over a large dataset of different X
CAMD Review 4
Figure 2: Example usage of group contribution methods
Original structure Group representation Number of occurrences
O F 1
O
C 2
F
F
Estimating properties
H
Group
n=[0,...,1,...,2,..., C 4
composition
4,...,1,...,1,...]
vector
Coefficient
c=[...,3.2,..., 2.4,..., O
vector − 1
0.6,...,1.2,...,2.3,...]
(example)
P = c n
Property g g
g 1
estimate X
=3.2(1) 2.4(2)+0.6(4)+1.2(1)+2.3(1)
−
wheref representssomefunctionoftheinnerproduct is shown below:
of the vectors c and n. These functions f appear in
manygroupcontributionmethodsandareimportant
P =f c n + c n + c n (3)
when predicted properties are not simple linear  g g g g g g
functions of the number of groups in a structure. gX∈F gX∈S gX∈T
 
Perhaps the most widely-used GC method in
Group contribution methods have also often
CAMD is that of Marrero and Gani [113, 114].
incorporated interaction terms [122, 123, 124, 125,
This method, like the method of Constantinou and
99, 140]. These are ways to include additional terms
Gani[32]thatpredatedit,providesanotherextension
to account for the simultaneous presence of two
to the general form of GC methods in that it
(same or different) groups in a particular structure.
introducesmultiplelevelsofgroupstobettercapture
For example, in predicting toxicity, one group in a
proximity effects, meaning the effect of two or more
molecule may lead to a simple metabolic pathway
groups which are close to one another in a molecular
and therefore make many structures containing that
structure. As many GC methods are limited to
group non-toxic. If that structure were also to have
groups of just a few atoms and bonds, many
another group which is normally quite toxic, a GC
cannotdifferentiatebetweenstructureswithdifferent
method without interaction terms may not predict
connectivity. The Marrero-Gani method, called the
toxicity well. This would be because both groups
GC+ method, uses as a first-order approximation a
would have an additive effect, meaning that there
normalgroupcontributionmethod,wherethegroups
would be one highly non-toxic contribution and one
belong to a set of primary groups F. An additional
highly toxic contribution. As a result, the molecule
setofgroupsScontainsslightlylargersub-structures.
may be predicted to have an average toxicity when,
Finally, a set of groups T accounts for large groups
in reality,the presence of the non-toxicgroupshould
andoverarchingmolecularstructuralfeatures. Unlike
outweighthe toxic group. Introducing aninteraction
groups in the set F, groups in the sets S and T
termforthesetwogroupsaccountsforthesituationof
are allowed to overlap with each other. Using the
their co-occurrence. In this example, this interaction
hierarchical depiction of molecules with the GC+
termwouldlikelyremovewhatevertoxicityvaluewas
method, a much clearer picture of a molecule is
predicted by the toxic group. For more discussion of
provided. The general form of the GC+ estimates
interaction terms in predicting toxicity, readers are
referred to [115]. Interaction terms generally take
the form
Ig,g′ =fI(ng,ng′) (4)
Austin, Sahinidis, and Trahan 5
where f usually represents multiplication, but can (V,E)todefineagraphG,withvertexsetV andedge
I
also represent other functions. Group contribution set E. Using this depiction of molecular structures,
methods can also include some idea of structural various properties of that graph, referred to here
features[122,123,124,125,113,114]. Theseaccount as topological indices, can be used as descriptors in
forlargereffects,typicallyatthemoleculescale,such QSPR models. More specifically, various topological
as aromatic ring substitution, cis/trans isomerism, indices are paired with regression coefficients and
aliphatic chain lengths, etc. These can typically be used to estimate properties in a similar way to GC
implemented as large groups but sometimes require methods.
special considerations. Table 1 provides several Topological indices (TIs) can take many forms.
properties typically used in CAMD problems and a They are defined as some function of the nodes and
few GC methods to estimate them. edgesinachemicalgraph,andonecaneasilyseethat
Strengths of GC methods. GC methods there are a large number of possible functions even
are useful in that they are very intuitive to use. justconsideringstandardmoleculargraphproperties
They represent a chemical structure in terms of like degree counts for nodes, connectivity, atomic
its functional components, very analogously to how types, etc. One of the first topological indices used
chemists compare and analyze structures. GC in chemical graph theory is the Wiener index [166].
methods are also able to easily represent a large and The Wiener index attempts to describe the total
diversechemicalspaceasthegroupscanbecombined distance between all atoms in the graph, as given
in many different ways to produce a large variety of by d(v,v′), the graph theoretic distance between all
different structures. This is especially useful from pairs of vertices v and v′. The Wiener index W(G)
a CAMD perspective. Finally, GC methods are is defined as:
easily translated into the mathematical formulations
ofCAMD problemsas the inclusionandcountofthe W(G)=1/2 d(v,v′) (5)
groups (the vector n) are easily represented in the v,v′
X
context of mathematical optimization.
WhiletheWienerindexdescribesagraphintermsof
Weaknesses of GC methods. There are a
itsdistances,anotherimportantconsiderationishow
few shortcomings of modern GC methods. One is
a graph is connected. To address this, an important
that many GC methods are unable to distinguish
classoftopologicalindicescalledconnectivityindices
isomers from one another. As isomers can have
(CIs)wasdeveloped. Connectivityindicesarewidely
very different properties, this represents a gap in
used in CAMD and have been shown to be useful
the predictive power of GC methods. We note that
in QSPR applications [44]. The first connectivity
some GC methods such as the GC+ methods are
indices were developed by Randi´c [142], who used
abletodistinguishmanyisomersduetotheinclusion
these indices to account for the degree of branching
of large groups. A second issue with GC methods
inalkanesandto modelenthalpyoffusionandvapor
is the lack of consistency in groups used to predict
pressure. Randi´c defined an edge index to be:
various properties. Though this has no major effect
estimating these properties for a given structure, it 1
CI (v,v′)= (6)
becomes problematic for mathematical formulations E
√δvδv′
of the CAMD problem. Finally, GC methods require
specifying the set of groups prior to regressing the where v and v′ are two connected vertices in the
GCcoefficients. ThoughmanyGCmethodsarequite chemical graph. This means that the atoms which
accurate,thereis noguaranteethatthe setofgroups correspond to v and v′ are connected by a chemical
used best captures the property they model. Using bond. Furthermore, δv and δv′ are the degrees of
different groups can sometimes drastically alter the nodes v and v′. In the study of Randi´c, these
predictive power of a GC model. degrees signified the number of σ bonds a particular
atom had to non-Hydrogen atoms (i.e., the number
atomicneighborsinthe hydrogen-suppressedgraph),
2.2 Topological indices
but they are sometimes defined differently for other
Chemical graph theory [15] is a field which became connectivity indices. The connectivity index of the
veryinfluentialinthe 1970’sandhassince been used entire molecule (graph) was then given by
to produce a large number of QSPRs. The basic
1
idea of chemical graph theory is that the atoms and 1χ= CI (v,v′)= (7)
E
bonds which constitute a molecule can be thought of {v,Xv′}∈E {v,Xv′}∈E √δvδv′
asnodesandedgesinagraph. Ingeneral,weuseG=
CAMD Review 6
Table 1: Sample of available GC methods for predicting various properties of pure compounds
Property GC methods
Aqueous solubility Marrero et al. [114], Klopman [100]
Joback and Reid [78], Stein et al. [157], Nannoolal et al. [122],
Boiling point
Marrero et al. [113]
Bond dissociation energy Benson et al. [10]
Jalowka et al. [76], Joback and Reid [78], Klincewicz et al. [99],
Critical pressure
Nannoolal et al. [123], Marrero et al. [113]
Jalowka et al. [76], Joback and Reid [78], Klincewicz et al. [99],
Critical temperature
Nannoolal et al. [123], Marrero et al. [113]
Critical volume Klincewicz et al. [99], Nannoolal et al. [123], Marrero et al. [113]
Cohen et al. [31], Benson [9], Domalski et al. [36], Roganov et
Enthalpy of formation
al. [144], Joback and Reid [78], Marrero et al. [113]
Enthalpy of fusion Joback and Reid [78], Marrero et al. [113]
Roganov et al. [144], Joback and Reid [78], Marrero et al. [113],
Enthalpy of vaporization
Ceriani et al. [24]
LC Martin et al. [115] (fathead minnow)
50
Melting point Joback and Reid [78], Marrero et al. [113]
Cohen et al. [31], Benson [9], Domalski et al. [36], Roganov et
Gibbs energy of formation
al. [144], Joback and Reid [78], Marrero et al. [113]
Benson et al. [11], Benson [9], Domalski et al. [36], Joback and
Heat capacity
Reid [78], Kolska´ et al. [103], Ceriani et al. [24]
Octanol/water partition
Marrero et al. [114], Platts et al. [140], Klopman et al. [101]
coefficient
Vapor pressure Nannoolal et al. [124]
Joback and Reid [78], Sastri et al. [149], Ceriani et al. [23], Cao
Viscosity
et al. [22], Nannoolal et al. [125]
where E is the edge set of the graph. A calculation Kier and Hall [87] developed additional connectivity
for the edge connectivity indices for a simple alkane indices to account for heteroatoms. These modified
is shown in Fig. 3. connectivity indices distinguished atoms by their
1χ is the so-called first-order Randi´c connectivity valence electrons, leading to a new value for vertex
index. An even simpler “connectivity index” exists degree, δV, where the superscript indicates that
which does not account for bonding at all. This valence is considered. For second period elements,
is called the zeroth-order connectivity index, and is Kier and Hall [87] define δV for a vertex v as
v
given below:
1 δV =ZV h (10)
0χ= (8) v v − v
√δ
v
vX∈V where ZvV indicates the number of valence electrons
Kier et al. [89], Hall et al. [65], and Murray et foratom/vertexv andh isthenumberofhydrogens
v
al. [121] were the first to apply these connectivity attached to v. For atoms in the third period and
indicesasdescriptorsinQSPRmodels. Thesemodels beyond, the main difference from this perspective is
are typically linear in the descriptors, but many the number of core electrons. These are accounted
variations exist. An important step forward came for in the following:
from Kier et al. [90] who introduced higher-order
connectivity indices. Ingeneral,these aredefined for δV = ZvV−hv (11)
an index of order i as: v Zv−ZvV−1
iχ= 1 (9) where Zv is the atomic number of atom v. These
δ modified δV values define analogous connectivity
{v1,v2X,...}∈VCi v v indices iχV.
qQ
The connectivity index χ can be thought to
where Vi is the setofallsets of i connectedvertices.
C
Austin, Sahinidis, and Trahan 7
Figure 3: Randi´c edge indices for a simple alkane
Structure Calculating CI (v,v′)
E
3
Atom Number (v) 1 2 3 4 5 6 7
e δ 1 3 1 4 1 1 1
1 v
2
7 CI for two edges
4 E
e2 1
CI (e1)=CI (1,2)= =0.577
E E
√1 3
1
×
6 1
CI (e2)=CI (4,6)= =0.500
E E
5 √4 1
×
primarily capture vertex adjacency and the local applications of QSPR development with topological
neighborhoodsofeveryatomin a molecule. As such, indices. We note that this table is inclined towards
it represents how atoms are connected and perhaps properties common in CAMD problems, so the
not the ensemble molecular structure and shape. properties listed may be more relevant to chemical
Another topological index often used in CAMD that engineering applications. However, we provide a
aims to address this is the so-called shape index, κ. few examples of the many applications of topological
The shape index accounts for features of the entire indices to modeling biological, environmental, and
molecular structure as functions of the underlying pharamacologicalproperties. There have been many
grapharchitectureandcountsofgraphsubstructures. efforts to build models related to pharmaceutical
The main parameter in the calculation of κ is the properties[7,45],soonlyasmallsubsetislistedhere.
number of paths in the chemical graph of a certain Oneadditionalinterestingapplicationofconnectivity
length. We define iP to be the number of paths of indices is in the prediction of coefficients for group
length i in a particular chemical graph. contributionmethodswherethegroupismissing(the
Like other topological indices, the shape index group is not in the descriptor space) [56].
maps this information onto a single value. There are Strengths of TIs. One of the main advantages
three often used iκ values which are defined as: of TIs is that they can discriminate between very
similar structures, often in cases where GC methods
2(1P )(1P )
1κ= max min (12) cannot (e.g., isomers). This provides a more holistic
(1P)2
picture of the molecule and can be very useful for
2(2P )(2P ) certaindesignproblems. Forexample, this mayhave
2κ= max min (13)
(2P)2 potential applications for CAMD in areas where a
4(3P )(3P ) structural feature of the to-be-designed compound is
3κ= max min (14) fixed a priori. Furthermore, since many TIs are a
(3P)2
function of the entire graph, TIs reflect the entire
withiP andiP representingthemaximumand nature of the molecular structure. This can have
max min
advantages over GC methods, which assume that
minimum possible number of paths of length i for a
hypothetical molecule with an equivalent number of each group provides a contribution independently
atoms. iP and iP can be easily derived from of other groups in the structure (this is offset
max min
somewhat by GC interaction functions). Finally,
graph theoretic arguments. The formal expressions
for these can be found in [63]. Several iκ values for TIs have been extensively applied to modeling
an example molecule are calculated in Fig. 4. pharmacologicalproperties. The quality and volume
of this literature means that TIs are very suitable to
In Table 2, we list a few references for QSPR
models using topological indices. There are many, many pharmaceutically relevant CAMD problems.
Weaknesses of TIs. Thoughtopologicalindices
many examples of such models, and we note that
this table is not meant to be exhaustive. For a have been widely applied to QSPR, there are only
more complete list of available topological-indices- limited examples in CAMD. Topological indices are
usually not as generally applicable as GC methods,
based QSPRs, the reader is directed to the book of
Devillers and Balaban [35]. meaning that TI-based QSPRs are often restricted
Table 2 provides some idea of the diversity of to a certain class of chemicals. For design purposes,
CAMD Review 8
i i
Figure 4: Getting P and calculating κ for an example molecule
Number of atoms Paths of length 1 Paths of length 2 Paths of length 3
0P: 6 1P: 5 2P: 6 3P: 4
2(1P )(1P ) 2(2P )(2P ) 4(3P )(3P )
1κ= max min 2κ= max min 3κ= max min
Calculating κ (1P)2 (2P)2 (3P)2
values 2(30)(5) 2(10)(4) 4(4)(3)
= =12 = =2.22 = =3
52 62 42
Other structures 0P 1P 2P 3P Other structures 0P 1P 2P 3P
6 6 6 6 6 5 7 3
6 5 4 3 6 7 11 13
this is problematic as it means that TI-based design atom in a molecule.
problems can only consider that particular subset AnalogouslytoTIs,SDsdefinethechemicalgraph
of the chemical search space. Furthermore, TIs to be G = (V,E). Standard SD methods also
represent graph-theoretic properties of the chemical incorporate node coloring for every node v via a
graph, and many of these properties are not always coloring function c (v) and the colors of each node
G
readily understandable from a chemical perspective C . This change is reflected in the slightly altered
v
(althoughRandi´c and Zupan [143] have offeredsome definitionofthe chemicalgraphasG=(V,E,C,c ).
G
interpretations of several TIs). Finally, TIs are These node colorings are intended to distinguish
more difficult to incorporate into CAMD than GC between different atoms as well as different types of
methods. We will discuss TI-based CAMD in a thesameatom. Forexample,itmaybe beneficialfor
subsequent section, but such CAMD problems can the model to differentiate between an oxygen with
sometimes face combinatorial difficulties and have two single bonds and an oxygenwith a double bond.
only been demonstrated thus far on small design Additionally, one may wantto distinguish hydrogens
problems. bywhatatomtheyareattachedto,aromaticcarbons
from non-aromaticones, atoms attachedto aromatic
rings, and many other chemical features. We note
2.3 Signature descriptors
that the colorings of the nodes are one of the
Inabroadsense,GCmethodscapturetheimportant only subjective parts of SD models, and different
subsetsofatomsinamoleculewhileTIsrelyonsome coloring schemes can have significant effects on the
function of the chemical graph. One QSPR method performance of the models.
which has been shown to capture aspects of both One important class of signature descriptors is
GC- and TI-based methods is signature descriptors known as atomic signatures. Given a certain atom
(SD). Signature descriptors are far younger than the in a chemical graph, its atomic signature represents
other methods discussed, originating in 2002 from alloftheatomsthatarewithinacertaindistance,or
Visco et al. [163] and 2003 from Faulon et al. [47]. height, from it. Varying the value of this distance
Like TIs, they conceive of chemical structures as the gives rise to different atomic signatures. In the
chemical graph. Rather than ascribe various values simplest case, with this distance set to 0, the atomic
to a complete molecular graph, SDs retain all of signature of an atom is simply that atom, colored in
the structuralandconnectivityinformationforevery keeping with the coloring definition. More formally,
foranatom(vertex)v, its atomicsignatureofheight
Austin, Sahinidis, and Trahan 9
Table2: SampleofavailableTI-basedmethodsforpredictingvariouspropertiesofpurecompounds
Property TI method
Anti-inflammatory activity Gupta et al. [61], Bajaj [5]
Aqueous diffusion coefficient Schramke [151]
Aqueous solubility Hall et al. [65], Katritzky et al. [85]
Biodegradability Boethling [12]
Blood-brain barrier partition
Rose et al. [145]
coefficients
Boiling point Hall et al. [65], Hosoya [73], Hall et al. [64], Galvez et al. [51]
Critical temperature Hall et al. [64]
Density Kier et al. [90], Estrada [43], Katritzky et al. [84]
Enthalpy of formation Mercader [118]
Enthalpy of fusion Gharagheizi et al. [59]
Enthalpy of vaporization Galvez et al. [51]
Flash point Patel et al. [135]
Heat capacity Yao et al. [169]
LC (fathead minnow) Hall et al. [66, 67], Basak et al. [8]
50
Melting point Katritzky et al. [84]
Nonspecific local anesthetic
Kier et al. [89], Katritzky et al. [84]
activity
Octanol/water partition
Murray et al. [121]
coefficient
π-electron energy (C-C bonds) Hosoya et al. [74]
Refractive index Katritzky et al. [84]
Vapor pressure Katritzky et al. [85]
Viscosity Kauffman et al. [86]
Water-air partition coefficient Katritzky et al. [85]
0, 0α is given by: increasing the i value is equivalent to adding a layer
to a breadth-first-search. A pictorial explanation
0α(v)=cG(v) (15) is provided in Fig. 5. In this example, we assume
that all atoms are colored by aromaticity and sp
Of course, these atomic signatures of height 0 are
hybridization and that hydrogens are colored by the
not a robust set of descriptors. Considering higher
atom they are attached to.
values for height provides a more detailed picture of
The atomic signatures can be thought of as
an atom’s environment. In general, for a height of i,
a descriptor space for a molecular structure. A
the atomic signature of height i 1 for a vertex v is
property, P, of a molecule can be estimated by all
≥
defined as:
of its atomic signatures of up to a particular height.
This QSPR has a familiar form:
iα(v)=G (Vi,Ei,Ci,c ) (16)
S G
P = c iα (d) (17)
where G defines a subgraph of G that contains all d G
S
vertices and bonds such that the distance between Xi dX∈Di
v and any vertex in G is at most i. Thus, an
S where d is the index of the set of all atomic
atomic signature of height one for a hypothetical signatures and Di is the set of atomic SDs of
atom v defines v and every atom bonded to v as
height i. c is a regression coefficient accounting
d
well as all connecting edges. A height two atomic
for the “contribution” of each atomic signature to
signature defines v, every atom bonded to v, and a certain property. iα (d) represents the number of
G
everyatombondedtothoseatomsbondedtov along
occurrences of atomic signature d. Using the atomic
with all necessary edges. Increasing the height of
signatures,signatures of the entire molecule can also
an atomic signature can thus be thought to add a
be generated.
layer of connected atoms. In graph-theoretic terms,
Amajoradvantageofsignaturedescriptorsisthat
CAMD Review 10
they can be manipulated via simple functions to estimatepropertiesfromastructure. Theapplication
representgroupsfromGCmethodsaswellasvarious of QSPR techniques in this direction—predicting
TIs. This means that the large amount of QSPRs properties from structures—defineswhatis knownas
derivedfromGCmethods andTI-basedmethods are the“forwardproblem,”andiswhatQSPRtechniques
accessible using SDs. For this reason, we do not are generally intended for. CAMD can broadly be
provide a table of QSPRs using SDs because they thought to consider the “reverse problem,” or the
can be—and often are—used to calculate properties problem of predicting structures from properties. At
via GC- and TI-based methods. A few examples of first glance, there is no immediately obvious way of
converting SDs to groups and TIs are given in [47]. relating properties to a specific molecular structure.
Strengths of SDs. One of the main advantages One issue is that there are so many structures to
of signature descriptors is that they have a small consider. A reasonable approach to the CAMD
inherent bias as compared to GC methods and problem should be able to consider a large diversity
TIs. The only bias introduced in these descriptors of structures without running into significant
is the choice of an atomic coloring scheme, or computational difficulties. Another issue involves
the choice of what defines a different type of the structural feasibility. Solutions to the CAMD
same atom. Furthermore, the inclusion of every problem must also be sensible molecular structures,
atomic signature in SD-based QSPRs means there meaning CAMD should produce structures that do
are no theoretical restrictions on the descriptor not violate any inherent laws of chemical bonding.
space defined for a molecular structure. There are A final issue involves consistency. All of the QSPRs
also a variety of more modern signature descriptors discussed (though non-overlapping GC methods are
capable of distinguishing stereoisomers. In the an exception) require that if certain features are
case of stereoisomers, certain SDs have far greater present, so must be other features. For example,
discriminative power than GC methods or TIs, it would be unreasonable to design a structure
which typically cannot differentiate stereoisomers. that has four paths of length three but no paths
Furthermore,theequivalenceofSDstomanyTIsand of length two. In the case of GC methods or
groups for GC methods means that SDs can be used SDs, an example erroneous solution would contain
directly with these TI and GC QSPR models. This one occurrence of the carboxylic acid (-C(=O)OH)
makes a large library of QSPR models accessible to group/signature and no occurrences of the carbonyl
SDs. (-C(=O)-) group/signature. Note that this example
Weaknesses of SDs. Many QSPRs using SDs assumes that overlap is allowed between these
use all of the available atomic signatures of up to groups/signatures(althoughthisisgenerallythecase
a certain height. This can quickly become an issue with signatures).
with the predictive power of SDs as models without Mathematical optimization is the key to address-
sufficient training data may be overfit. A second ing all of these issues. Before introducing the
concern with SDs is that the coloring scheme of the problem in a general optimization formulation, we
descriptors must be specified before the descriptors define a few important sets and variables. First,
are used for QSPRs and CAMD problems. It is we assume that we have a vector of properties p
likely that some coloring schemes provide better and a property value p for each property k. The
k
resultsthanothers,andthebestcoloringschememay vectornencapsulatesrelevantstructuralinformation
not always be easy to determine. Finally, atomic of the designed molecules and is dependent on
signature descriptors always discriminate between the type of QSPR chosen. In the case of GC
identical atoms in different structural environments. methods, the value n would represent the number
d
This may again lead to issues of overfitting or not ofoccurrencesof eachgroupd. For TIs, the n vector
capturingthe“true”chemicalbehaviorofthesystem may represent the number of topological features d
as sometimes it is better to model identical atoms (edges, paths of certain lengths, etc.) from which
withageneraldescriptorwhichis independentofthe TIs would be calculated. For SDs, this n vector
atom’s environment (although this can be captured usually representscounts of variousatomic signature
to some degree with SDs of lower heights). descriptors d. The function f then transforms this
structuralinformationintoapropertyestimateusing
the appropriate QSPR relationship. In general, the
3 CAMD as an optimization problem
The varioustypes ofstructure-propertyrelationships
discussed above can quickly and often accurately